Now. Old Swedish followed the traditional Germanic four-case system, and for the third person masculine singular it looked like this:

Nominative: han
Genitive: hans
Accusative: han
Dative: honom

In standard Swedish, the dative forms have everywhere crowded out the accusative forms and established themselves as general object forms. But in quite a lot of dialects the accusative lived on as an enclitic, so people would say “Jag såg’an” or “Jag såg’en” (I saw ‘im). It seems as if this usage not only has survived, but in fact expanded, so that “han” is not just viewed as an accusative enclitic, but in general is used by many language users in the object position. For me, it is fully grammatical to say “Han har inte jag sett på tio år” (I haven’t seen him in ten years). But no one will consider the use of “honom” in the object position to be strange, overly formal or anything like that, so don’t unlearn it.

A final aside: for the third person singular feminine, the Old Swedish paradigm was: hon – hennes – hana – henne. You sometimes hear “hana” be used in the object form in the form of an enclitic “‘na”, such as in “Jag såg’na” instead of “Jag såg henne”. This might be somewhat less common among younger language users, especially in urban areas, but far from extinct.

Both are pronounced ‘dom’ (don’t ask me why). You can pronounce them as ‘de’ and ‘dem’, but it sounds somewhat old-fashioned and pompous.

Since they are pronounced the same way, many swedes don’t know when to use which in writing. The most common rule is in fact to translate the sentence into english and if it translates to ‘them’ then it’s ‘dem’ otherwise ‘de’.

Another rule is to always write ‘dom’ which works in either case, but should be avoided in formal situations.

In the google fight:
Det är honom – 8 250
Det är han – 227 000

Ultimately, if the advice from above is that both are acceptable, then both are acceptable. I would use whichever you hear most often. You probably will end up using that which you hear most often anyway, whether you like it or not!

I saw him – I sag honom
He saw me – Han sag mig and so on.

Now I am confused!